1. André L. S. de Pinho*+ Harold J. Steudel* Søren Bisgaard#
Follow-up Experiments to Remove Confounding Between Location and Dispersion Effects in Unreplicated Two-Level Factorial Designs
André L. S. de Pinho*+
Harold J. Steudel*
Søren Bisgaard#
*Department of Industrial Engineering - University of Wisconsin-Madison
+Department of Statistics - Federal University of Rio Grande do Norte (UFRN) - Brazil
#Eugene M. Isenberg School of Management - University of Massachusetts, Amherst
Good afternoon. I would like to start by thanking every and each one of you for accepting my invitation to be a member in my prelim exam. I would like to give a special thank you to Prof. Steudel for his outstanding guidance throughout my research.
I will give now a little bit of background. Prof. Bisgaard is my co-advisor; he currently has a position in the School of Business at the Umass- Amherst. He helped us selected this topic and gave some initial research directions. Therefore, it is important to acknowledge his contribution to this work.

2. Outline Introduction Montgomery’s (1990) Injection Molding Experiment
Motivation
Montgomery’s (1990) Injection Molding Experiment
Research Proposal
Current Research Results
This is an overview of my presentation. We are going to start with an introduction that includes the motivation that led us to pursue this topic as well as the literature review., Next we will introduce an experimental data that illustrate the problem of confounding between location and dispersion effects. Then, we will make the link with our research proposal, establishing the results obtained so far. We will finish by summarizing our research contributions and presenting our research plan for the summer and fall semesters.

3. Introduction Motivation Current trend: Design for Six Sigma (DSS)
Ferocious competition in the market
High pressure for lowering cost, shortening time-to-market and increase reliability
Need to have faster, better and cheaper processes
Current trend: Design for Six Sigma (DSS)
Approach: Robust product design
Making products robust to process variability
DOE provides the means to achieve this goal
It is not new that we live in a very competitive world. Therefore, there is a high demand for lowering the cost, shortening the lead-time and improve the quality and reliability of products and services. It is fear to say, then, that there is a real need to make things faster, better and cheaper. In accordance with this features is the current quality trend known as Six-Sigma program. Embedded in the Six-Sigma program we have the Design for Six-Sigma, which tell us that we should spend a great deal of time and effort in the early stages of development of products and services, so that we avoid spending time correcting and making adjustments to problems that arise downstream.
One way to approach the DSS is to use robust design, and then make the product insensitive to process variability. The DOE certainly provides the means to achieve this goal.

4. Montgomery’s (1990) Injection Molding Experiment
fractional factorial design plus four center points with the objective of reducing the average parts shrinkage and also reducing the variability in shrinkage from run to run.
The factors studied
mold temperature (A), screw speed (B), holding time (C), gate size (D), cycle time (E), moisture content (F), and holding pressure (G).
The generators of the design were E = ABC, F = BCD, and G = ACD
We have a 27-3 resolution IV design with 16 runs. The response variable is the shrinkage, and we want to have the response as low as possible with low variability. Originally, the design also considers 4 center points, but we are not going to consider them because there is a possibility that the variance is not constant. In that case, the estimate of the variance from the center points could not be used to access the lack of fit. We have seven variables, 3 generators.

5. Injection Molding Experiment Data
i ↓ j → A 1 B 2 C 3 D 4 AB 5 AC 6 CG 7 AE 8 BD 9 AG 10 E 11
ABD 12 G 13 F 14 AF 15 Y 16 -1 32 60 26 34 37 52
And here we have our experimental data. I’d like to make a remark here! Since we have a fractional factorial design the choice to label the factors is not unique. We will use this notation throughout this analysis. Some time we will refer to the column number instead of the factor.

6. Probability Plot of Effects
We have two analyses in this plot. Montgomery identified A, B and AB as the active effects, whereas McGrath and Lin in addition those also included G and CG. McGrath and Lin refer to them as marginally significant relatively to the others.

7. Plausible Location Models
Two possible location models:
Montgomery’s Model
(M1) = A B AB
McGrath and Lin’s model
(M2) = A B AB –2.6875CG – G
So there are two possible location models. M1 with 4 parameters and M2 with 6 parameters.

8. Dispersion Effect Analysis
Box and Meyer Dispersion Effect Statistics
Dispersion effect
If we perform a d.e. analysis we find that M1 model has strong indication of a d.e. in C. Let’s recall that the interaction of the added l.e. in M2 is exactly C! So that rises a interesting question!

9. Conclusion Montgomery's Model (dispersion effect in C)
(M1) = A B AB
(dispersion effect in C)
McGrath and Lin’s Model
(M2) = A B AB –2.6875CG – G
(no dispersion effects [d.e.])
Which of the following models the most plausible one, or more appropriate to describe the shrinkage response?

10. Minimum Number of Trials
Montgomery’s (1990) injection molding
Addressed by McGrath (2001), 4 extra runs
The selection is done in such a way that A and B are fixed and each combination of the settings for columns 7 and 13 occurs
There are four sets of rows, (1, 5, 9, 13), (2, 6, 10, 14), (3, 7, 11, 15), and (4, 8, 12, 16). He selected (1, 5, 9, 13)
Going back to the injection molding experiment, McGrath determined the minimum number of additional runs to be used in a follow-up design when we have one d.e.
Remark: The result only tell us how many we should use, it doesn’t specify which one!
In this case we need only 4 extra runs. They should be selected in a way that A and B are in a fixed level and have a full factorial for factors C and G. There are four sets of trials that satisfy this condition. McGrath suggested (1,5,9,13).

11. Minimum Number of Trials Graphical Representation
A B
C G
- -
+ -
- +
+ +
32, 34 R = 2
60, 60 R = 0 B
Recommended runs for replication
4, 16 R = 12
15, 5 R = 10
Because it is near the optimum condition. We can see that the choice provides minimum shrinkage response. Our point, however, is that not necessarily this trials will present the best option among the existent ones. At this point, our primarily objective is to identify the factors and see how they affect the response variable, and not optimize process. We need, then, to develop a framework that allows us to make the selection according to some criterion. C
6, 8 R = 2
10, 12 R = 2 A

12. Research Proposal
Expanding Meyer, Steinberg and Box (1996) to accommodate the presence of d.e. in the models
3 - Sequential design method for discrimination among concurrent models
[Box and Hill (1967)]
1 - Bayesian method of finding active factors
in fractionated screening experiments
[Box and Meyer (1993)]
2 - Apply a suitable transformation to ensure constant variance
Our research proposal gives this possibility. We expanded the follow-up method of Meyer et al to accommodate the presence of dispersion effect in the models. There are three steps in our methodology. The first one is a bayesian procedure to identify location models. The second one is a suitable transformation to make the new-scaled response to have constant variance. And finally, the model discrimination criterion to suggest the follow-up trials. We will see in the next slides with more detail each step.

13. 1 - Bayesian Method of Finding Active Factors
Scenario
Fractionated Factorial Designs
Sparsity Principle Underlines the Process Being Studied
Allow the Inclusion of Non-Structured Models
Usually highly fractionated plans are used in screening designs. The underline assumption is that of sparsity principle. We also allow for the consideration of non-structured models, and that’s what makes our approach different from Box and Meyer (1993).

14. Bayesian Method of Finding Active Factors Cont.
Interpretation of the Posterior Probability
The first one can be regarded as a penalty for increasing the number of variables in the model Mi. The second component is nothing less than a measure of fit
After some algebra we have this expression for the posterior probability of Model i. At first sight it may appear a complex formula, but that is not quite true. We can view the posterior probability as the product of two parts. The first one is a penalty for increasing the number of parameters in the location model. The second one represents a measure of fit. Therefore, we will look for models with few parameters and a good fit, those will be the one with high posterior probability. We have a program in the appendix that calculates the posterior probability for the models. However, before we compute these probabilities we need to specify the value for gamma.

15. Finding Active Factors – Injection Molding Experiment
Using the injection mold experiment, the value of gamma that maximizes the unscaled posterior probability is 1.5. Applying the estimated gamma we can now computed the posterior probability of each factor. We can see that there is an agreement with Montgomery’s analysis. Only A and B are the active factors signaled by this method. However this was done without considering the non-structured model M2.
Considering structured models
Considering non-structured models
Marginal Posterior Probabilities – Pj

16. Model Discrimination (MD) Criterion OverviewX
Response M2 M1
Two Rival Models
Two Possible Models
(M1) and (M2) to describe a Response
We now give an overview of the discrimination criterion. We have two possible models, M1 and M2. For low values of X the two models a very similar to each other. Therefore, if we want to differentiate between the two we should observe high values of X, and then check which model is more appropriate. That’s the idea of the MD criterion, to search for a region where the rival models are the least similar to each other.

17. MD Criterion Cont. MD in the context of DOE:
The expression for the MD criterion, as we can see, depends on the posterior probabilities calculated in the previous step. Another important point is the fact that not only the distance between the response considering different models is presented, but also the precision of the response. Nonetheless, we can only use this method if we have constant variance. Therefore, we ought have some sort of transformation in the response to meet the requirements of the MD criterion.
Remark: Must have constant variance for all models considered!

18. Outlines of the Transformation Procedure
Use WLS of the expanded location model is in the sense of the Bergman and Hynén (1997) method of identifying dispersion effects
Once we have available the residuals from the expanded location model we can then calculate the ratio,
The expanded location model is the union of the set L and the two-way interaction between d and the elements of L. Once we have available the residuals from the expanded location model we can then calculate the ratio of the variances at the levels of the d.e.

19. Rearranged Covariance Matrix of Y}
d (-) }
Symmetric
d (+)

20. Transformation – Injection Molding Experiment
Montgomery’s (1990) Injection Molding Experiment
(M1) = A B AB. (d.e. in C)
(M2) = A B AB –2.6875CG – G. (no d.e.)
The minimum number of trials to resolve the confounding problem is four
The possible sets of four runs that can be used for the follow-up experiment are (1, 5, 9, 13), (2, 6, 10, 14), (3, 7, 11, 15), and (4, 8, 12, 16)
McGrath then suggested (1, 5, 9, 13) for replication because it is near the optimum condition.
Applying this idea into the injection molding experimental data we get the following. Let’s first recall what we have obtained thus far. Two models, one with d.e. in C and the other one with no dispersion effect. The follow-up experiments will have four runs. McGrath suggested (1,5,9,13) since it’s near the experimental optimum.

21. Finding the Expanded Model
The set of active location effects is L = {I, A, B, AB}
The set of dispersion effect is D = {C}
M1-expanded model is represented by the set = {I, A, B, C, AB, AC, BC, ABC}
(M1-expanded)
= A B – C AB – AC – BC ABC
The estimated weight is = 0.167
In this case we have three active effects, the d.e. is C. Then the expanded model is given by … Now, from the residuals we can calculate the estimated delta and apply the transformation.

22. MD Criterion – Injection Molding
MD criterion and the design points
32, 34 R = 2
4, 16 R = 12
60, 60 R = 0
6, 8 R = 2
10, 12 R = 2
15, 5 R = 10
60, 52 R = 8
26, 27 R = 11 A C B
Recommended
runs for replication
A B
C G
+ +
- -
+ -
- +
+ + MD
Design Points 4 8 12 16 1 5 9 13 2 6 10 14
9.1064 3 7 11 15
Surprisingly, McGrath’s suggestion for the follow-up was the least one to discriminate among the models. There is a great difference between the first choice, 53 and the last one, 7. We can observe that the region to run the follow-up actually provides maximum shrinkage response, and that’s is ok, since we are most interested in understand how the factors are influencing our response.
Remark: McGrath’s suggestion, (1, 5, 9, 13), was the second-best
discriminated follow-up design!

23. References
Bergman, B. and Hynén, A. (1997). “Dispersion Effects from Unreplicated Designs in the 2k-p Series”, Technometrics, 39, 2,
Box, G. E. P. and Hill, W. J. (1967). “Discrimination Among Mechanistic Models”, Technometrics, 9, 1,
Box, G. E. P. and Meyer, R. D. (1993). “Finding the Active Factors in Fractionated Screening Experiments”, Journal of Quality Technology, 25, 2,
McGrath, R. N. (2001). “Unreplicated Fractional Factorials: Two Location Effects or One Dispersion Effect?”, Joint Statistical Meetings (JSM) in Atlanta.
Meyer, R. D., Steinberg, D. M., and Box, G. E. P. (1996). “Follow-up Designs to Resolve Confounding in Multifactor Experiments”, Technometrics, 38, 4,
Montgomery, D. C. (1990). “Using Fractional Factorial Designs for Robust Process Development”, Quality Engineering, 3, 2,

24. Thank you for your time!